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- Burdakov, Oleg, 1953-, et al.
(författare)
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On a Reformulation of Mathematical Programs with Cardinality Constraints
- 2015
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Ingår i: Advances in Global Optimization. - Switzerland : Springer International Publishing. - 9783319083766 - 9783319083773 ; , s. 3-14
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Bokkapitel (refereegranskat)abstract
- Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.
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| 2. |
- Burdakov, Oleg, 1953-
(författare)
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On properties of Newton's method for smooth and nonsmooth equations
- 1995
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Ingår i: Recent Trends in Optimization Theory and Applications. - : World Scientific. - 9789810223823 - 9789812798862 ; , s. 17-24
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Variational inequalities, nonlinear programming, complementarity problems and other problems can be reduced to nonsmooth equations, for which some generalizations of Newton's method are known. The Newton path, as a natural generalization of the Newton direction, was suggested by D.Ralph for enlarging the convergence region (globalization) of Newton-Robinson's method in the nonsmooth case. We investigate some properties of both the Newton direction and the Newton path, which seem to be basic for various globalization strategies. In particular, a simple formula for the derivative of an arbitrary norm of residuals along the Newton direction,derived earlier by the author for the smooth equations, is generalizedhere for the derivative along the Newton path.
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